Sensitivity and Dynamic Range

The performance of RF receivers is characterized by many parameters. We study two, namely, sensitivity and dynamic range, here and defer the others to Chapter 3.

2.4.1. Sensitivity

The sensitivity is defined as the minimum signal level that a receiver can detect with “acceptable quality.” In the presence of excessive noise, the detected signal becomes unintelligible and carries little information. We define acceptable quality as sufficient signal-to-noise ratio, which itself depends on the type of modulation and the corruption (e.g., bit error rate) that the system can tolerate. Typical required SNR levels are in the range of 6 to 25 dB (Chapter 3).

In order to calculate the sensitivity, we write Equation 2.146, 2.147

where Psig denotes the input signal power and PRS the source resistance noise power, both per unit bandwidth. Do we express these quantities in V2/Hz or W/Hz? Since the input impedance of the receiver is typically matched to that of the antenna (Chapter 4), the antenna indeed delivers signal power and noise power to the receiver. For this reason, it is common to express both quantities in W/Hz (or dBm/Hz). It follows that

Equation 2.148

Since the overall signal power is distributed across a certain bandwidth, B, the two sides of (2.148) must be integrated over the bandwidth so as to obtain the total mean squared power. Assuming a flat spectrum for the signal and the noise, we have

Equation 2.149

Equation (2.149) expresses the sensitivity as the minimum input signal that yields a given value for the output SNR. Changing the notation slightly and expressing the quantities in dB or dBm, we have[22]

[22] Note that in conversion to dB or dBm, we take 10 log because these are power quantities. Equation 2.150

where Psen is the sensitivity and B is expressed in Hz. Note that (2.150) does not directly depend on the gain of the system. If the receiver is matched to the antenna, then from (2.91_{), PRS = kT = −174 dBm/Hz and}

Equation 2.151

Note that the sum of the first three terms is the total integrated noise of the system (sometimes called the “noise floor”). Example 2.25.

A GSM receiver requires a minimum SNR of 12 dB and has a channel bandwidth of 200 kHz. A wireless LAN receiver, on the other hand, specifies a minimum SNR of 23 dB and has a channel bandwidth of 20 MHz. Compare the

sensitivities of these two systems if both have an NF of 7 dB. Solution

For the GSM receiver, Psen = −102 dBm, whereas for the wireless LAN system, Psen = −71 dBm. Does this mean that the latter is inferior? No, the latter employs a much wider bandwidth and a more efficient modulation to accommodate a data rate of 54 Mb/s. The GSM system handles a data rate of only 270 kb/s. In other words, specifying the

sensitivity of a receiver without the data rate is not meaningful.

2.4.2. Dynamic Range

Dynamic range (DR) is loosely defined as the maximum input level that a receiver can “tolerate” divided by the minimum input level that it can detect (the sensitivity). This definition is quantified differently in different applications. For example, in analog circuits such as analog-to-digital converters, the DR is defined as the “full-scale” input level divided by the input level at which SNR = 1. The full scale is typically the input level beyond which a hard saturation occurs and can be easily determined by examining the circuit. In RF design, on the other hand, the situation is more complicated. Consider a simple common-source stage. How do we define the input “full scale” for such a circuit? Is there a particular input level beyond which the circuit becomes excessively nonlinear? We may view the 1-dB compression point as such a level. But, what if the circuit senses two interferers and suffers from intermodulation? In RF design, two definitions of DR have emerged. The first, simply called the dynamic range, refers to the maximum tolerable desired signal power divided by the minimum tolerable desired signal power (the sensitivity). Illustrated in Fig. 2.56(a), this DR is limited by compression at the upper end and noise at the lower end. For example, a cell phone coming close to a base station may receive a very large signal and must process it with acceptable distortion. In fact, the cell phone measures the signal strength and adjusts the receiver gain so as to avoid compression. Excluding interferers, this “compression-based” DR can exceed 100 dB because the upper end can be raised relatively easily.

Figure 2.56. Definitions of (a) DR and (b) SFDR. [View full size image]

The second type, called the “spurious-free dynamic range” (SFDR), represents limitations arising from both noise and interference. The lower end is still equal to the sensitivity, but the upper end is defined as the maximum input level in a two-tone test for which the third-order IM products do not exceed the integrated noise of the receiver. As shown in Fig. 2.56(b), two (modulated or unmodulated) tones having equal amplitudes are applied and their level is raised until the IM products reach the integrated

noise.[23] The ratio of the power of each tone to the sensitivity yields the SFDR. The SFDR represents the maximum relative level of interferers that a receiver can tolerate while producing an acceptable signal quality from a small input level.

[23] Note that the integrated noise is a single value (e.g., 100 μVrms), not a density.

Where should the various levels depicted in Fig. 2.56(b) be measured, at the input of the circuit or at its output? Since the IM components appear only at the output, the output port serves as a more natural candidate for such a measurement. In this case, the sensitivity—usually an input-referred quantity—must be scaled by the gain of the circuit so that it is referred to the output. Alternatively, the output IM magnitudes can be divided by the gain so that they are referred to the input. We follow the latter approach in our SFDR calculations.

To determine the upper end of the SFDR, we rewrite Eq. (2.56) as Equation 2.152

where, for the sake of brevity, we have denoted 20 log Ax as Px even though no actual power may be transferred at the input or output ports. Also, PIM,out represents the level of IM products at the output. If the circuit exhibits a gain of G (in dB), then we can refer the IM level to the input by writing PIM,in = PIM,out − G. Similarly, the input level of each tone is given by Pin = Pout − G. Thus, (2.152) reduces to

Equation 2.153, 2.154

and hence Equation 2.155

The upper end of the SFDR is that value of Pin which makes PIM,in equal to the integrated noise of the receiver: Equation 2.156

The SFDR is the difference (in dB) between Pin,max and the sensitivity: Equation 2.157, 2.158

For example, a GSM receiver with NF = 7 dB, PIIP3 = −15 dBm, and SNRmin = 12 dB achieves an SFDR of 54 dB, a substantially lower value than the dynamic range in the absence of interferers.

Example 2.26.

The upper end of the dynamic range is limited by intermodulation in the presence of two interferers or desensitization in the presence of one interferer. Compare these two cases and determine which one is more restrictive.

Solution

We must compare the upper end expressed by Eq. (2.156) with the 1-dB compression point: Equation 2.159

Equation 2.160

and hence Equation 2.161

Since the right-hand side represents the receiver noise floor, we expect it to be much lower than the left-hand side. In fact, even for an extremely wideband channel of B = 1 GHz and NF = 10 dB, the right-hand side is equal to −74 dBm, whereas, with a typical PIIP3 of −10 to −25 dBm, the left-hand side still remains higher. It is therefore plausible to conclude that

Equation 2.162

It follows that the maximum tolerable level in a two-tone test is quite lower than that in a compression test, i.e., corruption by intermodulation between two interferers is much greater than compression due to one. The SFDR is therefore a more stringent characteristic of the system than the compression-based dynamic range.